Introduction to MATLAB

1. Introduction to MATLAB ES 156 Signals and Systems 2007 Harvard SEAS Prepared by Jiajun Gu

2. Outline • Introduction and where to get MATLAB • Data structure: matrices, vectors and operations • Basic line plots • File I/O

3. Where to get MATLAB • FAS computing: • Download from http://fas.harvard.edu/computing/software/ • You must be on FAS network to use MATLAB • HSEAS IT • Maxwell Dworkin Rooms G107-G111 • Mathworks: • Student version is affordable and complete.

4. What is MATLAB • High level language for technical computing • Stands for MATrix LABoratory • Everything is a matrix - easy to do linear algebra

5. The MATLAB System • Development Environment • Mathematical Function Library • MATLAB language • Application Programming Language (not discussed today)

6. MATLAB Desktop Menu and toolbar Workspace History Command

7. >> A = [16 3; 5 10] A = 16 3 5 10 Matrices & Vectors • All (almost) entities in MATLAB are matrices • Easy to define: • Use ‘,’ or ‘ ’ to separate row elements -- use ‘;’ to separate rows

8. Matrices & Vectors - II • Order of Matrix - • m=no. of rows, n=no. of columns • Vectors - special case • n = 1 column vector • m = 1 row vector

9. Matrix: >> A=[1 2; 3 4]; >> A' ans = 1 3 2 4 Vector : >> a=[1 2 3]; >> a' 1 2 3 Creating Vectors and Matrices >> A = [16 3; 5 10] A = 16 3 5 10 >> B = [3 4 5 6 7 8] B = 3 4 5 6 7 8 • Define • Transpose

10. Creating Vectors Create vector with equally spaced intervals >> x=0:0.5:pi x = 0 0.5000 1.0000 1.5000 2.0000 2.5000 3.0000 Create vector with n equally spaced intervals >> x=linspace(0, pi, 7) x = 0 0.5236 1.0472 1.5708 2.0944 2.6180 3.1416 Equal spaced intervals in logarithm space >> x=logspace(1,2,7) x = 10.0000 14.6780 21.5443 … 68.1292 100.0000 Note: MATLAB uses pi to represent , uses i or j to represent imaginary unit

11. Creating Matrices • zeros(m, n):matrix with all zeros • ones(m, n): matrix with all ones. • eye(m, n): the identity matrix • rand(m, n): uniformly distributed random • randn(m, n): normally distributed random • magic(m): square matrix whose elements have the same sum, along the row, column and diagonal. • pascal(m) : Pascal matrix.

12. Matrix operations • ^:exponentiation • *: multiplication • /: division • \: left division. The operation A\Bis effectively the same as INV(A)*B, although left division is calculated differently and is much quicker. • +: addition • -: subtraction

13. Array Operations • Evaluated element by element .' : array transpose (non-conjugated transpose) .^ : array power .* : array multiplication ./ : array division • Very different from Matrix operations >> A=[1 2;3 4]; >> B=[5 6;7 8]; >> A*B 19 22 43 50 But: >> A.*B 5 12 21 32

14. Some Built-in functions • mean(A):mean value of a vector • max(A), min (A): maximum and minimum. • sum(A): summation. • sort(A): sorted vector • median(A): median value • std(A): standard deviation. • det(A) : determinant of a square matrix • dot(a,b): dot product of two vectors • Cross(a,b): cross product of two vectors • Inv(A): Inverse of a matrix A

15. Indexing Matrices n A = 0.9501 0.6068 0.4231 0.2311 0.4860 0.2774 m Given the matrix: Then: A(1,2) = 0.6068 A(3) = 0.6068 A(:,1) = [0.9501 0.2311 ] A(1,2:3)=[0.6068 0.4231] 1:m

16. Adding Elements to a Vector or a Matrix >> C=[1 2; 3 4] C= 1 2 3 4 >> C(3,:)=[5 6]; C= 1 2 3 4 5 6 >> D=linspace(4,12,3); >> E=[C D’] E= 1 2 4 3 4 8 5 6 12 >> A=1:3 A= 1 2 3 >> A(4:6)=5:2:9 A= 1 2 3 5 7 9 >> B=1:2 B= 1 2 >> B(5)=7; B= 1 2 0 0 7

17. Graphics - 2D Plots plot(xdata, ydata, ‘marker_style’); For example: Gives: >> x=-5:0.1:5; >> sqr=x.^2; >> pl1=plot(x, sqr, 'r:s');

18. Graphics - Overlay Plots Use hold on for overlaying graphs So the following: Gives: >> hold on; >> cub=x.^3; >> pl2=plot(x, cub,‘b-o');

19. Graphics - Annotation Use title, xlabel, ylabel and legend for annotation >> title('Demo plot'); >> xlabel('X Axis'); >> ylabel('Y Axis'); >> legend([pl1, pl2], 'x^2', 'x^3');

20. Graphics - Annotation

21. Graphics-Stem() • stem()is to plot discrete sequence data • The usage of stem() is very similar to plot() >> n=-10:10; >> f=stem(n,cos(n*pi/4)) >> title('cos(n\pi/4)') >> xlabel('n')

22. subplots • Use subplots to divide a plotting window into several panes. >> x=0:0.1:10; >> f=figure; >> f1=subplot(1,2,1); >> plot(x,cos(x),'r'); >> grid on; >> title('Cosine') >> f2=subplot(1,2,2); >> plot(x,sin(x),'d'); >> grid on; >> title('Sine');

23. Save plots • Use saveas(h,'filename.ext') to save a figure to a file. Useful extension types: bmp: Windows bitmap emf: Enhanced metafile eps: EPS Level 1 fig: MATLAB figure jpg: JPEG image m: MATLAB M-file tif: TIFF image, compressed >> f=figure; >> x=-5:0.1:5; >> h=plot(x,cos(2*x+pi/3)); >> title('Figure 1'); >> xlabel('x'); >> saveas(h,'figure1.fig') >> saveas(h,'figure1.eps')

24. Workspace • Matlab remembers old commands • And variables as well • Each Function maintains its own scope • The keyword clearremoves all variables from workspace • The keyword who lists the variables

25. File I/O • Matlab has a native file format to save and load workspaces. Use keywords load and save. • In addition MATLAB knows a large number of popular formats. Type “help fileformats” for a listing. • In addition MATLAB supports ‘C’ style low level file I/O. Type “help fprintf” for more information.

26. Practice Problems • Plot the following signals in linear scale • Plot the following signals, use log scale for y-axis • Plot the real part and imaginary part of the following signal • For the signal in previous question, plot its phase and magnitude